Modeling 9/11 disaster networks using degree sequence models Miruna Petrescu-Prahova, University of Washington Michael Schweinberger, Pennsylvania State University Duy Quang Vu, Pennsylvania State University June 3, 2011 Miruna Petrescu-Prahova, University of Washington Disaster Networks , Michael Schweinberger, Pennsylvania State University
9/11 organizational collaboration network Response to World Trade Center attacks Data on interorganizational collaboration during the first 12 days after September 11, 2001 was collected from newspapers, field documents, and interviews Organizational attributes: type of organization (e.g., non-profit, governmental, etc.) and scale of the organization (e.g., local, state, etc.) Result: network of collaborations among 717 organizations 226 isolates 10 small components of sizes 2—10 one large component of size 463 Miruna Petrescu-Prahova, University of Washington Disaster Networks , Michael Schweinberger, Pennsylvania State University
9/11 organizational collaboration network local state national international collective government non-profit profit Miruna Petrescu-Prahova, University of Washington Disaster Networks , Michael Schweinberger, Pennsylvania State University
9/11 organizational collaboration network: challenges Degree distribution: small number of high-degree nodes (coordinators) and large number of low-degree nodes ⇒ communication (coordination) depends on small number of nodes. 0.5 0.4 0.3 0.2 0.1 0.0 0 20 40 60 80 Miruna Petrescu-Prahova, University of Washington Disaster Networks , Michael Schweinberger, Pennsylvania State University
Modeling Fundamental feature of collaboration network: sequence of degrees g 1 ( y ) , . . . , g n ( y ), where g i ( y ) = � n j � = i y ij . Maximum entropy principle (Jaynes, 1957a,b) suggests following family of exponential-family models: � n � � P θ ( Y = y ) = exp θ i g i ( Y ) − A ( θ ) i =1 n � , = exp ( θ i + θ j ) y ij − A ( θ ) i < j where n � A ( θ ) = log [1 + exp ( θ i + θ j )] . i < j Miruna Petrescu-Prahova, University of Washington Disaster Networks , Michael Schweinberger, Pennsylvania State University
Modeling and learning Modeling: Simple starting point. Motivated by data. Dyad-independence exponential-family random graph model. Natural relative of p 1 model of Holland and Leinhardt (1981), the first exponential-family random graph model. Can be extended to include covariates. Learning: Bayesian approach with truncated Dirichlet process prior and hyper-prior distribution: Bayesian approach to stochastic block model. Posterior distribution approximated by variational methods as well as MCMC. Miruna Petrescu-Prahova, University of Washington Disaster Networks , Michael Schweinberger, Pennsylvania State University
Outline of analyses Fit ERGMs that include both dyad-independent and dyad-dependent terms. Select number of blocks using variational method Fit stochastic block models that include dyad-independent terms. Compare the goodness-of-fit of the stochastic block models and ERGMs. Analyze composition of blocks Miruna Petrescu-Prahova, University of Washington Disaster Networks , Michael Schweinberger, Pennsylvania State University
ERGMs Terms: edges, nodemix, 2-star, gwdegree. Covariates: organizational type (4 categories) and organizational scale of operations (4 categories). Goodness-of-fit: compare the properties of the observed network with same properties of simulated networks. M1: edges, nodemix M2: edges, nodemix, 2-star M3: edges, nodemix, gwdegree Miruna Petrescu-Prahova, University of Washington Disaster Networks , Michael Schweinberger, Pennsylvania State University
Stochastic block models Terms: edges and nodemix. Covariates: organizational type (4 categories) and organizational scale of operations (4 categories). First step: determining the best fit in terms of number of blocks using variational EM. Miruna Petrescu-Prahova, University of Washington Disaster Networks , Michael Schweinberger, Pennsylvania State University
Selecting the number of blocks ● ● ● ● ● ● −6400 ● −6600 log P(Y = y) −6800 −7000 ● −7200 ● 2 4 6 8 10 #blocks M4: stochastic block model with 4 blocks Primary, secondary and tertiary hubs and near-isolates (Butts et al, 2007). Miruna Petrescu-Prahova, University of Washington Disaster Networks , Michael Schweinberger, Pennsylvania State University
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